neuper@37906
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(* differentiation over the reals
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neuper@37906
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author: Walther Neuper
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neuper@37906
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000516
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neuper@37906
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*)
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neuper@37906
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wneuper@59424
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theory Diff imports Calculus Trig LogExp Rational Root Poly Base_Tools begin
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neuper@37906
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wneuper@59472
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ML \<open>
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neuper@37993
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@{term "sin x"}
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wneuper@59472
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\<close>
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neuper@37993
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neuper@37906
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consts
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neuper@37906
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neuper@37906
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d_d :: "[real, real]=> real"
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wneuper@59551
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neuper@37906
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(*descriptions in the related problems*)
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neuper@37993
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derivativeEq :: "bool => una"
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neuper@37906
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neuper@37906
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(*predicates*)
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neuper@37906
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primed :: "'a => 'a" (*"primed A" -> "A'"*)
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neuper@37906
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walther@60242
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(*the CAS-commands, eg. "Diff (2*x \<up> 3, x)",
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neuper@37906
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"Differentiate (A = s * (a - s), s)"*)
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neuper@37906
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Diff :: "[real * real] => real"
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neuper@37906
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Differentiate :: "[bool * real] => bool"
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neuper@37906
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wneuper@59551
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(*subproblem-name*)
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wneuper@59484
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differentiate :: "[char list * char list list * char list, real, real] => real"
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neuper@37906
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("(differentiate (_)/ (_ _ ))" 9)
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neuper@37906
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wneuper@59472
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text \<open>a variant of the derivatives defintion:
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neuper@37906
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neuper@37954
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d_d :: "(real => real) => (real => real)"
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neuper@37954
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neuper@37954
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advantages:
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neuper@37954
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(1) no variable 'bdv' on the meta-level required
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neuper@37954
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(2) chain_rule "d_d (%x. (u (v x))) = (%x. (d_d u)) (v x) * d_d v"
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neuper@37954
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(3) and no specialized chain-rules required like
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neuper@37954
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diff_sin_chain "d_d bdv (sin u) = cos u * d_d bdv u"
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neuper@37954
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neuper@37954
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disadvantage: d_d (%x. 1 + x^2) = ... differs from high-school notation
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wneuper@59472
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\<close>
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neuper@37954
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neuper@52148
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axiomatization where (*stated as axioms, todo: prove as theorems
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neuper@37906
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'bdv' is a constant on the meta-level *)
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neuper@52148
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diff_const: "[| Not (bdv occurs_in a) |] ==> d_d bdv a = 0" and
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neuper@52148
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diff_var: "d_d bdv bdv = 1" and
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neuper@37983
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diff_prod_const:"[| Not (bdv occurs_in u) |] ==>
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neuper@52148
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d_d bdv (u * v) = u * d_d bdv v" and
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neuper@37906
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neuper@52148
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diff_sum: "d_d bdv (u + v) = d_d bdv u + d_d bdv v" and
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neuper@52148
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diff_dif: "d_d bdv (u - v) = d_d bdv u - d_d bdv v" and
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neuper@52148
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diff_prod: "d_d bdv (u * v) = d_d bdv u * v + u * d_d bdv v" and
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neuper@37983
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diff_quot: "Not (v = 0) ==> (d_d bdv (u / v) =
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walther@60242
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(d_d bdv u * v - u * d_d bdv v) / v \<up> 2)" and
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neuper@37906
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neuper@52148
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diff_sin: "d_d bdv (sin bdv) = cos bdv" and
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neuper@52148
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diff_sin_chain: "d_d bdv (sin u) = cos u * d_d bdv u" and
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neuper@52148
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diff_cos: "d_d bdv (cos bdv) = - sin bdv" and
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neuper@52148
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diff_cos_chain: "d_d bdv (cos u) = - sin u * d_d bdv u" and
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walther@60242
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diff_pow: "d_d bdv (bdv \<up> n) = n * (bdv \<up> (n - 1))" and
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walther@60242
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diff_pow_chain: "d_d bdv (u \<up> n) = n * (u \<up> (n - 1)) * d_d bdv u" and
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neuper@52148
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diff_ln: "d_d bdv (ln bdv) = 1 / bdv" and
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neuper@52148
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diff_ln_chain: "d_d bdv (ln u) = d_d bdv u / u" and
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neuper@52148
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diff_exp: "d_d bdv (exp bdv) = exp bdv" and
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neuper@52148
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diff_exp_chain: "d_d bdv (exp u) = exp u * d_d x u" and
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neuper@37906
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(*
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neuper@37906
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diff_sqrt "d_d bdv (sqrt bdv) = 1 / (2 * sqrt bdv)"
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neuper@37906
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diff_sqrt_chain"d_d bdv (sqrt u) = d_d bdv u / (2 * sqrt u)"
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neuper@37906
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*)
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neuper@37906
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(*...*)
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neuper@37906
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neuper@37983
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frac_conv: "[| bdv occurs_in b; 0 < n |] ==>
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walther@60242
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a / (b \<up> n) = a * b \<up> (-n)" and
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walther@60242
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frac_sym_conv: "n < 0 ==> a * b \<up> n = a / b \<up> (-n)" and
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neuper@37906
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walther@60242
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sqrt_conv_bdv: "sqrt bdv = bdv \<up> (1 / 2)" and
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walther@60242
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sqrt_conv_bdv_n: "sqrt (bdv \<up> n) = bdv \<up> (n / 2)" and
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walther@60269
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(*Ambiguous input\<^here> produces 3 parse trees -----------------------------\\*)
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walther@60242
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sqrt_conv: "bdv occurs_in u ==> sqrt u = u \<up> (1 / 2)" and
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walther@60269
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(*Ambiguous input\<^here> produces 3 parse trees -----------------------------//*)
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walther@60242
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sqrt_sym_conv: "u \<up> (a / 2) = sqrt (u \<up> a)" and
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neuper@37906
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walther@60242
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root_conv: "bdv occurs_in u ==> nroot n u = u \<up> (1 / n)" and
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walther@60242
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root_sym_conv: "u \<up> (a / b) = nroot b (u \<up> a)" and
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neuper@37906
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walther@60242
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realpow_pow_bdv: "(bdv \<up> b) \<up> c = bdv \<up> (b * c)"
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neuper@37906
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wneuper@59472
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ML \<open>
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neuper@37972
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val thy = @{theory};
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neuper@37972
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neuper@37954
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(** eval functions **)
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neuper@37954
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neuper@37954
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fun primed (Const (id, T)) = Const (id ^ "'", T)
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neuper@37954
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| primed (Free (id, T)) = Free (id ^ "'", T)
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walther@59962
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| primed t = raise ERROR ("primed called with arg = '"^ UnparseC.term t ^"'");
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neuper@37954
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neuper@37954
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(*("primed", ("Diff.primed", eval_primed "#primed"))*)
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walther@60335
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fun eval_primed _ _ (p as (Const (\<^const_name>\<open>Diff.primed\<close>,_) $ t)) _ =
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walther@59868
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SOME ((UnparseC.term p) ^ " = " ^ UnparseC.term (primed t),
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wneuper@59390
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HOLogic.Trueprop $ (TermC.mk_equality (p, primed t)))
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neuper@37954
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| eval_primed _ _ _ _ = NONE;
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wneuper@59472
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\<close>
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wenzelm@60313
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wenzelm@60313
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calculation primed = \<open>eval_primed "#primed"\<close>
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wenzelm@60313
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wneuper@59472
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ML \<open>
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neuper@37954
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(** rulesets **)
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neuper@37954
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neuper@37954
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(*.converts a term such that differentiation works optimally.*)
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neuper@37954
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val diff_conv =
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walther@59851
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Rule_Def.Repeat {id="diff_conv",
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neuper@37954
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preconds = [],
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neuper@37954
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rew_ord = ("termlessI",termlessI),
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walther@59852
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erls = Rule_Set.append_rules "erls_diff_conv" Rule_Set.empty
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wenzelm@60294
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[\<^rule_eval>\<open>Prog_Expr.occurs_in\<close> (Prog_Expr.eval_occurs_in ""),
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wenzelm@60297
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\<^rule_thm>\<open>not_true\<close>,
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wenzelm@60297
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\<^rule_thm>\<open>not_false\<close>,
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wenzelm@60294
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\<^rule_eval>\<open>less\<close> (Prog_Expr.eval_equ "#less_"),
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wenzelm@60297
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\<^rule_thm>\<open>and_true\<close>,
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wenzelm@60297
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\<^rule_thm>\<open>and_false\<close>
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neuper@37954
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],
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walther@59851
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srls = Rule_Set.Empty, calc = [], errpatts = [],
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neuper@42449
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rules =
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wenzelm@60297
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[\<^rule_thm>\<open>frac_conv\<close>,
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walther@60242
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(*"?bdv occurs_in ?b \<Longrightarrow> 0 < ?n \<Longrightarrow> ?a / ?b \<up> ?n = ?a * ?b \<up> - ?n"*)
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wenzelm@60297
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\<^rule_thm>\<open>sqrt_conv_bdv\<close>,
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walther@60242
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(*"sqrt ?bdv = ?bdv \<up> (1 / 2)"*)
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wenzelm@60297
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\<^rule_thm>\<open>sqrt_conv_bdv_n\<close>,
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walther@60242
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(*"sqrt (?bdv \<up> ?n) = ?bdv \<up> (?n / 2)"*)
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wenzelm@60297
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\<^rule_thm>\<open>sqrt_conv\<close>,
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walther@60242
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(*"?bdv occurs_in ?u \<Longrightarrow> sqrt ?u = ?u \<up> (1 / 2)"*)
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wenzelm@60297
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\<^rule_thm>\<open>root_conv\<close>,
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walther@60242
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(*"?bdv occurs_in ?u \<Longrightarrow> nroot ?n ?u = ?u \<up> (1 / ?n)"*)
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wenzelm@60297
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\<^rule_thm>\<open>realpow_pow_bdv\<close>,
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walther@60242
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(* "(?bdv \<up> ?b) \<up> ?c = ?bdv \<up> (?b * ?c)"*)
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wenzelm@60294
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\<^rule_eval>\<open>times\<close> (**)(eval_binop "#mult_"),
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wenzelm@60297
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\<^rule_thm>\<open>rat_mult\<close>,
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neuper@42449
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(*a / b * (c / d) = a * c / (b * d)*)
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wenzelm@60297
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\<^rule_thm>\<open>times_divide_eq_right\<close>,
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neuper@42449
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(*?x * (?y / ?z) = ?x * ?y / ?z*)
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wenzelm@60297
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\<^rule_thm>\<open>times_divide_eq_left\<close>
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neuper@42449
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(*?y / ?z * ?x = ?y * ?x / ?z*)
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neuper@37954
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],
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walther@59878
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scr = Rule.Empty_Prog};
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wneuper@59472
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\<close>
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wneuper@59472
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ML \<open>
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neuper@37954
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(*.beautifies a term after differentiation.*)
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neuper@37954
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val diff_sym_conv =
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walther@59851
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Rule_Def.Repeat {id="diff_sym_conv",
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neuper@37954
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preconds = [],
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neuper@37954
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rew_ord = ("termlessI",termlessI),
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walther@59852
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erls = Rule_Set.append_rules "erls_diff_sym_conv" Rule_Set.empty
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walther@60331
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[\<^rule_eval>\<open>less\<close> (Prog_Expr.eval_equ "#less_"),
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walther@60331
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walther@60331
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Rule.Eval ("Prog_Expr.matches", Prog_Expr.eval_matches "#matches_"),
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walther@60330
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Rule.Eval ("Prog_Expr.is_atom", Prog_Expr.eval_is_atom "#is_atom_"),
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walther@60330
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Rule.Eval ("Orderings.ord_class.less", Prog_Expr.eval_equ "#less_"),
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walther@60337
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Rule.Thm ("not_false", @{thm not_false}),
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walther@60337
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Rule.Thm ("not_true", @{thm not_true})],
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walther@59851
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srls = Rule_Set.Empty, calc = [], errpatts = [],
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wenzelm@60297
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rules = [\<^rule_thm>\<open>frac_sym_conv\<close>,
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wenzelm@60297
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\<^rule_thm>\<open>sqrt_sym_conv\<close>,
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wenzelm@60297
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\<^rule_thm>\<open>root_sym_conv\<close>,
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wenzelm@60296
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\<^rule_thm_sym>\<open>real_mult_minus1\<close>,
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wenzelm@60296
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(*- ?z = "-1 * ?z"*)
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wenzelm@60297
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\<^rule_thm>\<open>rat_mult\<close>,
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walther@60278
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(*a / b * (c / d) = a * c / (b * d)*)
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wenzelm@60297
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\<^rule_thm>\<open>times_divide_eq_right\<close>,
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walther@60278
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(*?x * (?y / ?z) = ?x * ?y / ?z*)
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wenzelm@60297
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\<^rule_thm>\<open>times_divide_eq_left\<close>,
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walther@60278
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(*?y / ?z * ?x = ?y * ?x / ?z*)
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wenzelm@60294
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\<^rule_eval>\<open>times\<close> (**)(eval_binop "#mult_")
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walther@60278
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],
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walther@59878
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scr = Rule.Empty_Prog};
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neuper@37954
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neuper@37954
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(*..*)
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neuper@37954
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val srls_diff =
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walther@59851
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Rule_Def.Repeat {id="srls_differentiate..",
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neuper@37954
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preconds = [],
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neuper@37954
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rew_ord = ("termlessI",termlessI),
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walther@59852
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erls = Rule_Set.empty,
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walther@59851
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srls = Rule_Set.Empty, calc = [], errpatts = [],
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wenzelm@60294
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rules = [\<^rule_eval>\<open>Prog_Expr.lhs\<close> (Prog_Expr.eval_lhs "eval_lhs_"),
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wenzelm@60294
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\<^rule_eval>\<open>Prog_Expr.rhs\<close> (Prog_Expr.eval_rhs "eval_rhs_"),
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wenzelm@60294
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\<^rule_eval>\<open>Diff.primed\<close> (eval_primed "Diff.primed")
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neuper@37954
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],
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walther@59878
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scr = Rule.Empty_Prog};
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wneuper@59472
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\<close>
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wneuper@59472
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ML \<open>
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neuper@37954
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(*..*)
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neuper@37954
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val erls_diff =
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walther@59852
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Rule_Set.append_rules "erls_differentiate.." Rule_Set.empty
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wenzelm@60297
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[\<^rule_thm>\<open>not_true\<close>,
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wenzelm@60297
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\<^rule_thm>\<open>not_false\<close>,
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neuper@37954
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wenzelm@60294
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\<^rule_eval>\<open>Prog_Expr.ident\<close> (Prog_Expr.eval_ident "#ident_"),
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wenzelm@60294
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\<^rule_eval>\<open>Prog_Expr.is_atom\<close> (Prog_Expr.eval_is_atom "#is_atom_"),
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wenzelm@60294
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\<^rule_eval>\<open>Prog_Expr.occurs_in\<close> (Prog_Expr.eval_occurs_in ""),
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wenzelm@60294
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\<^rule_eval>\<open>Prog_Expr.is_const\<close> (Prog_Expr.eval_const "#is_const_")
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neuper@37954
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];
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neuper@37954
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neuper@37954
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(*.rules for differentiation, _no_ simplification.*)
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neuper@37954
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val diff_rules =
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walther@59851
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Rule_Def.Repeat {id="diff_rules", preconds = [], rew_ord = ("termlessI",termlessI),
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walther@59851
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erls = erls_diff, srls = Rule_Set.Empty, calc = [], errpatts = [],
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wenzelm@60297
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rules = [\<^rule_thm>\<open>diff_sum\<close>,
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wenzelm@60297
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\<^rule_thm>\<open>diff_dif\<close>,
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wenzelm@60297
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\<^rule_thm>\<open>diff_prod_const\<close>,
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wenzelm@60297
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\<^rule_thm>\<open>diff_prod\<close>,
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wenzelm@60297
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\<^rule_thm>\<open>diff_quot\<close>,
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wenzelm@60297
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212 |
\<^rule_thm>\<open>diff_sin\<close>,
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wenzelm@60297
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\<^rule_thm>\<open>diff_sin_chain\<close>,
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wenzelm@60297
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\<^rule_thm>\<open>diff_cos\<close>,
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wenzelm@60297
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\<^rule_thm>\<open>diff_cos_chain\<close>,
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wenzelm@60297
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\<^rule_thm>\<open>diff_pow\<close>,
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wenzelm@60297
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\<^rule_thm>\<open>diff_pow_chain\<close>,
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wenzelm@60297
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218 |
\<^rule_thm>\<open>diff_ln\<close>,
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wenzelm@60297
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\<^rule_thm>\<open>diff_ln_chain\<close>,
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wenzelm@60297
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\<^rule_thm>\<open>diff_exp\<close>,
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wenzelm@60297
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\<^rule_thm>\<open>diff_exp_chain\<close>,
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neuper@37954
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222 |
(*
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wenzelm@60297
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223 |
\<^rule_thm>\<open>diff_sqrt\<close>,
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wenzelm@60297
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\<^rule_thm>\<open>diff_sqrt_chain\<close>,
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neuper@37954
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*)
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wenzelm@60297
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\<^rule_thm>\<open>diff_const\<close>,
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wenzelm@60297
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\<^rule_thm>\<open>diff_var\<close>
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neuper@37954
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],
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walther@59878
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229 |
scr = Rule.Empty_Prog};
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wneuper@59472
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\<close>
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wneuper@59472
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ML \<open>
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neuper@37954
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232 |
(*.normalisation for checking user-input.*)
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neuper@37954
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233 |
val norm_diff =
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walther@59851
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234 |
Rule_Def.Repeat
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neuper@42458
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235 |
{id="norm_diff", preconds = [], rew_ord = ("termlessI",termlessI),
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walther@59851
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236 |
erls = Rule_Set.Empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
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wneuper@59416
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rules = [Rule.Rls_ diff_rules, Rule.Rls_ norm_Poly ],
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walther@59878
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scr = Rule.Empty_Prog};
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wneuper@59472
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\<close>
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wenzelm@60289
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240 |
rule_set_knowledge
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wenzelm@60286
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241 |
erls_diff = \<open>prep_rls' erls_diff\<close> and
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wenzelm@60286
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diff_rules = \<open>prep_rls' diff_rules\<close> and
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wenzelm@60286
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norm_diff = \<open>prep_rls' norm_diff\<close> and
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wenzelm@60286
|
244 |
diff_conv = \<open>prep_rls' diff_conv\<close> and
|
wenzelm@60286
|
245 |
diff_sym_conv = \<open>prep_rls' diff_sym_conv\<close>
|
s1210629013@55363
|
246 |
|
wenzelm@60306
|
247 |
|
wenzelm@60306
|
248 |
(** problems **)
|
wenzelm@60306
|
249 |
|
wenzelm@60306
|
250 |
problem pbl_fun : "function" = \<open>Rule_Set.empty\<close>
|
wenzelm@60306
|
251 |
|
wenzelm@60306
|
252 |
problem pbl_fun_deriv : "derivative_of/function" =
|
wenzelm@60306
|
253 |
\<open>Rule_Set.append_rules "empty" Rule_Set.empty []\<close>
|
wenzelm@60306
|
254 |
Method: "diff/differentiate_on_R" "diff/after_simplification"
|
wenzelm@60306
|
255 |
CAS: "Diff (f_f, v_v)"
|
wenzelm@60306
|
256 |
Given: "functionTerm f_f" "differentiateFor v_v"
|
wenzelm@60306
|
257 |
Find: "derivative f_f'"
|
wenzelm@60306
|
258 |
|
wenzelm@60306
|
259 |
problem pbl_fun_deriv_nam :
|
wenzelm@60306
|
260 |
"named/derivative_of/function" (*here "named" is used differently from Integration"*) =
|
wenzelm@60306
|
261 |
\<open>Rule_Set.append_rules "empty" Rule_Set.empty []\<close>
|
wenzelm@60306
|
262 |
Method: "diff/differentiate_equality"
|
wenzelm@60306
|
263 |
CAS: "Differentiate (f_f, v_v)"
|
wenzelm@60306
|
264 |
Given: "functionEq f_f" "differentiateFor v_v"
|
wenzelm@60306
|
265 |
Find: "derivativeEq f_f'"
|
s1210629013@55380
|
266 |
|
wneuper@59472
|
267 |
ML \<open>
|
neuper@37954
|
268 |
(** CAS-commands **)
|
neuper@37954
|
269 |
|
neuper@37954
|
270 |
(*.handle cas-input like "Diff (a * x^3 + b, x)".*)
|
neuper@37954
|
271 |
(* val (t, pairl) = strip_comb (str2term "Diff (a * x^3 + b, x)");
|
wenzelm@60309
|
272 |
val [Const (\<^const_name>\<open>Pair\<close>, _) $ t $ bdv] = pairl;
|
neuper@37954
|
273 |
*)
|
wenzelm@60309
|
274 |
fun argl2dtss [Const (\<^const_name>\<open>Pair\<close>, _) $ t $ bdv] =
|
walther@60339
|
275 |
[(TermC.parseNEW'' \<^theory> "functionTerm", [t]),
|
walther@60339
|
276 |
(TermC.parseNEW'' \<^theory> "differentiateFor", [bdv]),
|
walther@60339
|
277 |
(TermC.parseNEW'' \<^theory> "derivative",
|
walther@60339
|
278 |
[TermC.parseNEW'' \<^theory> "f_f'"])
|
neuper@37954
|
279 |
]
|
walther@59962
|
280 |
| argl2dtss _ = raise ERROR "Diff.ML: wrong argument for argl2dtss";
|
wneuper@59472
|
281 |
\<close>
|
wenzelm@60303
|
282 |
|
wenzelm@60303
|
283 |
method met_diff : "diff" =
|
wenzelm@60303
|
284 |
\<open>{rew_ord'="tless_true",rls'=Atools_erls,calc = [], srls = Rule_Set.empty, prls=Rule_Set.empty,
|
wenzelm@60303
|
285 |
crls = Atools_erls, errpats = [], nrls = norm_diff}\<close>
|
wneuper@59545
|
286 |
|
wneuper@59504
|
287 |
partial_function (tailrec) differentiate_on_R :: "real \<Rightarrow> real \<Rightarrow> real"
|
wneuper@59504
|
288 |
where
|
walther@59635
|
289 |
"differentiate_on_R f_f v_v = (
|
walther@59635
|
290 |
let
|
walther@59635
|
291 |
f_f' = Take (d_d v_v f_f)
|
walther@59635
|
292 |
in (
|
walther@59637
|
293 |
(Try (Rewrite_Set_Inst [(''bdv'',v_v)] ''diff_conv'')) #> (
|
walther@59635
|
294 |
Repeat (
|
walther@59635
|
295 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_sum'')) Or
|
walther@59635
|
296 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_prod_const'')) Or
|
walther@59635
|
297 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_prod'')) Or
|
walther@59635
|
298 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_quot'')) Or
|
walther@59635
|
299 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_sin'')) Or
|
walther@59635
|
300 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_sin_chain'')) Or
|
walther@59635
|
301 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_cos'')) Or
|
walther@59635
|
302 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_cos_chain'')) Or
|
walther@59635
|
303 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_pow'')) Or
|
walther@59635
|
304 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_pow_chain'')) Or
|
walther@59635
|
305 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_ln'')) Or
|
walther@59635
|
306 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_ln_chain'')) Or
|
walther@59635
|
307 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_exp'')) Or
|
walther@59635
|
308 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_exp_chain'')) Or
|
walther@59635
|
309 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_const'')) Or
|
walther@59635
|
310 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_var'')) Or
|
walther@59637
|
311 |
(Repeat (Rewrite_Set ''make_polynomial'')))) #> (
|
walther@59635
|
312 |
Try (Rewrite_Set_Inst [(''bdv'',v_v)] ''diff_sym_conv''))
|
walther@59635
|
313 |
) f_f')"
|
wenzelm@60303
|
314 |
|
wenzelm@60303
|
315 |
method met_diff_onR : "diff/differentiate_on_R" =
|
wenzelm@60303
|
316 |
\<open>{rew_ord'="tless_true", rls' = erls_diff, calc = [], srls = Rule_Set.empty, prls=Rule_Set.empty,
|
wenzelm@60303
|
317 |
crls = Atools_erls, errpats = [], nrls = norm_diff}\<close>
|
wenzelm@60303
|
318 |
Program: differentiate_on_R.simps
|
wenzelm@60303
|
319 |
Given: "functionTerm f_f" "differentiateFor v_v"
|
wenzelm@60303
|
320 |
Find: "derivative f_f'"
|
wneuper@59545
|
321 |
|
wneuper@59504
|
322 |
partial_function (tailrec) differentiateX :: "real \<Rightarrow> real \<Rightarrow> real"
|
wneuper@59504
|
323 |
where
|
walther@59635
|
324 |
"differentiateX f_f v_v = (
|
walther@59635
|
325 |
let
|
walther@59635
|
326 |
f_f' = Take (d_d v_v f_f)
|
walther@59635
|
327 |
in (
|
walther@59635
|
328 |
Repeat (
|
walther@59635
|
329 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_sum'')) Or
|
walther@59635
|
330 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_prod_const'' )) Or
|
walther@59635
|
331 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_prod'')) Or
|
walther@59635
|
332 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_quot'')) Or
|
walther@59635
|
333 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_sin'')) Or
|
walther@59635
|
334 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_sin_chain'')) Or
|
walther@59635
|
335 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_cos'')) Or
|
walther@59635
|
336 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_cos_chain'')) Or
|
walther@59635
|
337 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_pow'')) Or
|
walther@59635
|
338 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_pow_chain'')) Or
|
walther@59635
|
339 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_ln'')) Or
|
walther@59635
|
340 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_ln_chain'')) Or
|
walther@59635
|
341 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_exp'')) Or
|
walther@59635
|
342 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_exp_chain'')) Or
|
walther@59635
|
343 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_const'')) Or
|
walther@59635
|
344 |
(Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_var'')) Or
|
walther@59635
|
345 |
(Repeat (Rewrite_Set ''make_polynomial'')))
|
walther@59635
|
346 |
) f_f')"
|
wenzelm@60303
|
347 |
|
wenzelm@60303
|
348 |
method met_diff_simpl : "diff/diff_simpl" =
|
wenzelm@60303
|
349 |
\<open>{rew_ord'="tless_true", rls' = erls_diff, calc = [], srls = Rule_Set.empty, prls=Rule_Set.empty,
|
wenzelm@60303
|
350 |
crls = Atools_erls, errpats = [], nrls = norm_diff}\<close>
|
wenzelm@60303
|
351 |
Program: differentiateX.simps
|
wenzelm@60303
|
352 |
Given: "functionTerm f_f" " differentiateFor v_v"
|
wenzelm@60303
|
353 |
Find: "derivative f_f'"
|
wneuper@59545
|
354 |
|
wneuper@59504
|
355 |
partial_function (tailrec) differentiate_equality :: "bool \<Rightarrow> real \<Rightarrow> bool"
|
wneuper@59504
|
356 |
where
|
walther@59635
|
357 |
"differentiate_equality f_f v_v = (
|
walther@59635
|
358 |
let
|
walther@59635
|
359 |
f_f' = Take ((primed (lhs f_f)) = d_d v_v (rhs f_f))
|
walther@59635
|
360 |
in (
|
walther@59637
|
361 |
(Try (Rewrite_Set_Inst [(''bdv'',v_v)] ''diff_conv'' )) #> (
|
walther@59635
|
362 |
Repeat (
|
walther@59635
|
363 |
(Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_sum'')) Or
|
walther@59635
|
364 |
(Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_dif'' )) Or
|
walther@59635
|
365 |
(Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_prod_const'')) Or
|
walther@59635
|
366 |
(Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_prod'')) Or
|
walther@59635
|
367 |
(Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_quot'')) Or
|
walther@59635
|
368 |
(Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_sin'')) Or
|
walther@59635
|
369 |
(Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_sin_chain'')) Or
|
walther@59635
|
370 |
(Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_cos'')) Or
|
walther@59635
|
371 |
(Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_cos_chain'')) Or
|
walther@59635
|
372 |
(Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_pow'')) Or
|
walther@59635
|
373 |
(Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_pow_chain'')) Or
|
walther@59635
|
374 |
(Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_ln'')) Or
|
walther@59635
|
375 |
(Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_ln_chain'')) Or
|
walther@59635
|
376 |
(Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_exp'')) Or
|
walther@59635
|
377 |
(Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_exp_chain'')) Or
|
walther@59635
|
378 |
(Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_const'')) Or
|
walther@59635
|
379 |
(Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_var'')) Or
|
walther@59637
|
380 |
(Repeat (Rewrite_Set ''make_polynomial'')))) #> (
|
walther@59635
|
381 |
Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''diff_sym_conv'' ))
|
walther@59635
|
382 |
) f_f')"
|
wenzelm@60303
|
383 |
|
wenzelm@60303
|
384 |
method met_diff_equ : "diff/differentiate_equality" =
|
wenzelm@60303
|
385 |
\<open>{rew_ord'="tless_true", rls' = erls_diff, calc = [], srls = srls_diff, prls=Rule_Set.empty,
|
wenzelm@60303
|
386 |
crls=Atools_erls, errpats = [], nrls = norm_diff}\<close>
|
wenzelm@60303
|
387 |
Program: differentiate_equality.simps
|
wenzelm@60303
|
388 |
Given: "functionEq f_f" "differentiateFor v_v"
|
wenzelm@60303
|
389 |
Find: "derivativeEq f_f'"
|
wneuper@59545
|
390 |
|
wneuper@59504
|
391 |
partial_function (tailrec) simplify_derivative :: "real \<Rightarrow> real \<Rightarrow> real"
|
wneuper@59504
|
392 |
where
|
walther@59635
|
393 |
"simplify_derivative term bound_variable = (
|
walther@59635
|
394 |
let
|
walther@59634
|
395 |
term' = Take (d_d bound_variable term)
|
walther@59635
|
396 |
in (
|
walther@59637
|
397 |
(Try (Rewrite_Set ''norm_Rational'')) #>
|
walther@59637
|
398 |
(Try (Rewrite_Set_Inst [(''bdv'', bound_variable)] ''diff_conv'')) #>
|
walther@59637
|
399 |
(Try (Rewrite_Set_Inst [(''bdv'', bound_variable)] ''norm_diff'')) #>
|
walther@59637
|
400 |
(Try (Rewrite_Set_Inst [(''bdv'', bound_variable)] ''diff_sym_conv'')) #>
|
walther@59635
|
401 |
(Try (Rewrite_Set ''norm_Rational''))
|
walther@59635
|
402 |
) term')"
|
walther@59634
|
403 |
|
wenzelm@60303
|
404 |
method met_diff_after_simp : "diff/after_simplification" =
|
wenzelm@60303
|
405 |
\<open>{rew_ord'="tless_true", rls' = Rule_Set.empty, calc = [], srls = Rule_Set.empty, prls=Rule_Set.empty,
|
wenzelm@60303
|
406 |
crls=Atools_erls, errpats = [], nrls = norm_Rational}\<close>
|
wenzelm@60303
|
407 |
Program: simplify_derivative.simps
|
wenzelm@60303
|
408 |
Given: "functionTerm term" "differentiateFor bound_variable"
|
wenzelm@60303
|
409 |
Find: "derivative term'"
|
wenzelm@60303
|
410 |
|
wenzelm@60314
|
411 |
cas Diff = \<open>argl2dtss\<close>
|
wenzelm@60314
|
412 |
Problem: "derivative_of/function"
|
wenzelm@60314
|
413 |
|
wneuper@59472
|
414 |
ML \<open>
|
neuper@37954
|
415 |
|
neuper@37954
|
416 |
(*.handle cas-input like "Differentiate (A = s * (a - s), s)".*)
|
neuper@37954
|
417 |
(* val (t, pairl) = strip_comb (str2term "Differentiate (A = s * (a - s), s)");
|
wenzelm@60309
|
418 |
val [Const (\<^const_name>\<open>Pair\<close>, _) $ t $ bdv] = pairl;
|
neuper@37954
|
419 |
*)
|
wenzelm@60309
|
420 |
fun argl2dtss [Const (\<^const_name>\<open>Pair\<close>, _) $ t $ bdv] =
|
walther@60339
|
421 |
[(TermC.parseNEW'' \<^theory> "functionEq", [t]),
|
walther@60339
|
422 |
(TermC.parseNEW'' \<^theory> "differentiateFor", [bdv]),
|
walther@60339
|
423 |
(TermC.parseNEW'' \<^theory> "derivativeEq",
|
walther@60339
|
424 |
[TermC.parseNEW'' \<^theory> "f_f'::bool"])
|
neuper@37954
|
425 |
]
|
walther@59962
|
426 |
| argl2dtss _ = raise ERROR "Diff.ML: wrong argument for argl2dtss";
|
wneuper@59472
|
427 |
\<close>
|
wenzelm@60314
|
428 |
cas Differentiate = \<open>argl2dtss\<close>
|
wenzelm@60314
|
429 |
Problem: "named/derivative_of/function"
|
wenzelm@60314
|
430 |
ML \<open>
|
walther@60278
|
431 |
\<close> ML \<open>
|
walther@60278
|
432 |
\<close>
|
neuper@37906
|
433 |
end
|